Annals of Mathematics, 150 (1999), 353–356
Correction to
Connecting invariant manifolds
and the solution of the C  stability
and Ω-stability conjectures for flows
By Shuhei Hayashi
There is a gap in the proof of Lemma VII.4 in [1]. We present an alternative proof of Theorem B (C 1 Ω-stable vector fields satisfy Axiom A) in
[1]. The novel and essential part in the proof of the stability and Ω-stability
conjectures for flows is the connecting lemma introduced in [1]. A mistake in
the proof of the last conjecture was pointed out to me by Toyoshiba [5], who
later also provided an independent proof of it, again based on the connecting
lemma and previous arguments by Mañé and Palis.
The crucial step to the proof of Theorem B is the separation of singularities from periodic orbits ([1, Corollary III]) by the C 1 connecting lemma
([1, Theorem A]). After the separation, the proof proceeds based on Mañé’s
theorems used in [3] and we still rely on Palis’s argument in [4], proving first
the density of Axiom A diffeomorphisms in the set of C 1 Ω-stable ones to then
show that every C 1 Ω-stable diffeomorphism satisfies Axiom A.
1 (M ) be the set of C 1 Ω-stable vector fields on a compact smooth
Let GΩ
1 (M ). As in [1], we
boundaryless manifold M with the C 1 topology and X ∈ GΩ
prove the hyperbolicity of Per(X) (= Ω(X) − Sing(X)) by induction. In fact,
S
we prove that P j (X) is hyperbolic assuming that j−1
i=0 P i (X) is hyperbolic for
some 1 ≤ j ≤ dimM − 1, where P i (X) is the closure of the set of periodic
points with index i (dimension of the stable subspace), which is enough to
1 (M ), we can
conclude that X satisfies Axiom A. For a dense subset of GΩ
use the statement of [1, Lemma VII.4] by an already classic argument on
set-valued functions of C 1 vector fields. In fact, there is a residual subset
1 (M )) in which the closure of
of the set of C 1 vector fields (therefore of GΩ
the set of hyperbolic periodic points of saddle type moves continuously with
respect to vector fields (see for instance the proof of [1, Corollary II] for this
kind of argument). Therefore, as proved in [1], we get the density of Axiom
1 (M ). Then, by Ω-conjugacy, we see that Ω(X) can be
A vector fields in GΩ
decomposed into a finite union of disjoint compact invariant sets which are
isolated and transitive. Moreover, Palis’s argument ([4]) for flows shows that
each component is homogeneous in the sense that the index of every periodic
354
SHUHEI HAYASHI
point in it is the same. Thus, the proof of Theorem B is reduced to proving
the following claim:
Claim. Every homogeneous component of P j (X) is hyperbolic.
Let G 1 (M ) be the interior of the set of C 1 vector fields on M , with the
topology, such that all periodic orbits and singularities are hyperbolic.
1 (M ) ⊂ G 1 (M ). Denote by LX , t ∈ R the linear Poincaré flow of
Then GΩ
t
1
X ∈ G (M ) on N ∗ (see [1, p. 126] for the definition). As in [1, p. 131], let
N ∗ |P j (X) = Ej ⊕ Fj be the dominated splitting such that
C1
X
kLX
m |Ej (y)k · kL−m |Fj (Xm (y))k ≤ λ
(1)
for all y ∈ P j (X) with m ∈ Z+ and 0 < λ < 1 given by [1, Lemma VII.1],
which is the continuous extension of hyperbolic splittings of periodic orbits of
index j with respect to LX
t . To prove the Claim, it is enough to show that
Ej is contracting by the following lemma proved in [3, Theorem II.1], which is
Lemma VII.5 in [1] and written for this setting:
Lemma 1 (Mañé). Let Λ be a compact invariant set of X ∈ G 1 (M ) such
that Λ ∩ Sing(X) = ∅ and Ω(X|Λ) = Λ. Suppose that N ∗ |Λ = E ⊕ F is a
dominated splitting such that the dimension of the subspaces E(y), y ∈ Λ is
constant,
X
kLX
m |E(y)k · kL−m |F (Xm (y))k ≤ λ
for all y ∈ Λ, and
lim inf n→+∞
n
1X
log kLX
−m |F (Xm` (x))k ≤ log λ
n `=1
holds for a dense set of points x ∈ Λ, where m ∈ Z+ and 0 < λ < 1 are given in
(1). Then, if E is contracting, F is expanding (and therefore Λ is hyperbolic).
P
Let (X) be the set of “strongly closable points” given in [1, Lemma
VII.6 (Ergodic Closing Lemma for time-one maps)] and originally introduced
by Mañé in [2]. We shall need the following lemma:
P
Lemma 2. Let X ∈ G 1 (M ). If x ∈ (X) and OX (x) ∩ Sing(X) = ∅,
then OX (x) contains a hyperbolic set, where OX (x) = {Xt (x) : t ∈ R}.
P
Proof. We can suppose that x ∈ (X) − Per(X). Let Un , n ≥ 0, be a
P
basis of neighborhoods of X. Then, by the definition of (X) ([1, p. 132];
see also [2, p. 506]), there exists {tn > 0 : n ≥ 0} with limn→+∞ tn = +∞,
X n ∈ Un and yn ∈ Per(X n ) having period Tn such that {Xt (x) : 0 ≤ t ≤ tn }
can be approximated by {Xtn (yn ) : 0 ≤ t ≤ Tn } for large n. Without loss of
CORRECTION TO: CONNECTING INVARIANT MANIFOLDS
355
generality we may assume that the index of the X n -periodic point yn is the
same for all n ≥ 0 (by taking a subsequence if necessary). Then, by [1, Lemma
VII.1], the following properties hold for all n ≥ 0 with Tn ≥ m:
n
n
s
X
u
n
kLX
m |En (y)k · kL−m |En (Xm (y))k ≤ λ
for all y ∈ {Xtn (yn ) : 0 ≤ t ≤ Tn }, and
[Tn /m]
Y
n
u
n
[Tn /m]
kLX
,
−m |En (Xm` (yn ))k ≤ Kλ
`=1
n
over the X n where Ens ⊕ Enu is the hyperbolic splitting with respect to LX
t
periodic orbit with yn . Note that the dimension of En (y) is constant, and
the angle between Ens (y) and Enu (y) is uniformly bounded away from 0 by [2,
Lemma II.9]. Then, defining the splitting E ⊕ F over OX (x) by accumulation
P
of {Ens ⊕Enu : n ≥ 0} (see the definition of (X) again), we have, by continuity,
the following properties:
X
kLX
m |E(y)k · kL−m |F (Xm (y))k ≤ λ
for all y ∈ OX (x), and
lim inf n→+∞
n
1X
log kLX
−m |F (Xm` (x))k ≤ log λ.
n `=1
It is well known that the dominated splitting E ⊕ F over OX (x) can
e ⊕ Fe over OX (x) with the same m ∈ Z+ and
be continuously extended to E
0 < λ < 1. Hence, we can apply Lemma 1 to Λ = OX (x). If OX (x) is not
e is not contracting, then, as in [1, p. 132], there exists
hyperbolic; that is, E
P
p ∈ OX (x) ∩ (X) such that
(2)
X
1 n−1
e
log kLX
m |E(Xm` (p))k ≥ 0.
n→+∞ n
`=0
lim
When p ∈ Per(X), OX (p) is a hyperbolic set, we may assume that p ∈
/
Per(X). Then, we can continue this argument for OX (p) instead of OX (x).
As observed in [1, pp. 132–133], the index i0 of periodic point created by the
e If i0 = 0, then, by the
Ergodic Closing Lemma from OX (p) is less than dim E.
same argument as in the proof of the finiteness of periodic orbits in P 0 (M ), p
cannot be reccurent. Therefore, we may suppose that OX (p) has a contracting
subbundle (by continuing this further if necessary) and therefore, by Lemma
1, OX (p) is hyperbolic, proving Lemma 2.
e of
Now let us prove the Claim. Assume that a homogeneous component Λ
P
e satisfying property
P j (X) is not hyperbolic. Then, we can find p ∈ (X) ∩ Λ
356
SHUHEI HAYASHI
e replaced by Ej in (1). Lemma 2 implies that OX (p) contains a
(2) with E
hyperbolic set Λ, and, as in the proof of Lemma 2, the index of any periodic
e
point in Λ is less than j. This contradicts the homogeneity of Λ.
School of Commerce, Waseda University, Shinjuku, Tokyo, Japan
E-mail address: shuhei@mn.waseda.ac.jp
References
[1] S. Hayashi, Connecting invariant manifolds and the solution of the C 1 stability and
Ω-stability conjectures for flows, Ann. of Math. 145 (1997), 81–137.
[2] R. Mañé, An ergodic closing lemma, Ann. of Math. 116 (1982), 503–540.
[3]
, A proof of the C 1 stability conjecture, Publ. Math. I.H.E.S. 66 (1988), 161–210.
[4] J. Palis, On the C 1 Ω-stability conjecture, Publ. Math. I.H.E.S. 66 (1988), 211–215.
[5] H. Toyoshiba, private communication.
(Received November 16, 1998)